How to Solve a System of Equations on the TI-84 Plus

{“appState”:{“pageLoadApiCallsStatus”:true},”articleState”:{“article”:{“headers”:{“creationTime”:”2016-03-26T13:59:00+00:00″,”modifiedTime”:”2016-03-26T13:59:00+00:00″,”timestamp”:”2022-09-14T18:12:56+00:00″},”data”:{“breadcrumbs”:[{“name”:”Technology”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33512″},”slug”:”technology”,”categoryId”:33512},{“name”:”Electronics”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33543″},”slug”:”electronics”,”categoryId”:33543},{“name”:”Graphing Calculators”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33551″},”slug”:”graphing-calculators”,”categoryId”:33551}],”title”:”How to Solve a System of Equations on the TI-84 Plus”,”strippedTitle”:”how to solve a system of equations on the ti-84 plus”,”slug”:”how-to-solve-a-system-of-equations-on-the-ti-84-plus”,”canonicalUrl”:””,”seo”:{“metaDescription”:”Matrices are the perfect tool for solving systems of equations (the larger the better). Fortunately, you can work with matrices on your TI-84 Plus. All you need”,”noIndex”:0,”noFollow”:0},”content”:”Matrices are the perfect tool for solving systems of equations (the larger the better). Fortunately, you can work with matrices on your TI-84 Plus. All you need to do is decide which method you want to use. \n<h2 id=\”tab1\” >A–1*B method of solving a system of equations</h2>\nWhat do the A and B represent? The letters A and B are capitalized because they refer to matrices. Specifically, A is the coefficient matrix and B is the constant matrix. In addition, X is the variable matrix. No matter which method you use, it’s important to be able to convert back and forth from a system of equations to matrix form.\n<img src=\”https://www.dummies.com/wp-content/uploads/400124.image0.jpg\” width=\”110\” height=\”64\” alt=\”image0.jpg\”/>\n<img src=\”https://www.dummies.com/wp-content/uploads/400125.image1.jpg\” width=\”487\” height=\”357\” alt=\”image1.jpg\”/>\nHere’s a short explanation of where this method comes from. Any system of equations can be written as the matrix equation, A * X = B. By pre-multiplying each side of the equation by A–1 and simplifying, you get the equation X = A–1 * B. \nUsing your calculator to find A–1 * B is a piece of cake. Just follow these steps:\n<ol class=\”level-one\”>\n <li>Enter the coefficient matrix, A.\nPress [ALPHA][ZOOM] to create a matrix from scratch or press [2nd][x–1] to access a stored matrix. See the first screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400126.image2.jpg\” width=\”535\” height=\”160\” alt=\”image2.jpg\”/>\n </li>\n <li>Press [x–1] to find the inverse of matrix A.\nSee the second screen.\n </li>\n <li>Enter the constant matrix, B.\n </li>\n <li>Press [ENTER] to evaluate the variable matrix, X.\nThe variable matrix indicates the solutions: x = 5, y = 0, and z = 1. See the third screen.\n </li>\n</ol>\nIf the determinant of matrix A is zero, you get the ERROR: SINGULAR MATRIX error message. This means that the system of equations has either no solution or infinite solutions.\n<h2 id=\”tab2\” >Augmenting matrices method to solve a system of equations</h2>\nAugmenting two matrices enables you to append one matrix to another matrix. Both matrices must be defined and have the same number of rows. Use the system of equations to augment the coefficient matrix and the constant matrix.\n<img src=\”https://www.dummies.com/wp-content/uploads/400127.image3.jpg\” width=\”110\” height=\”64\” alt=\”image3.jpg\”/>\nTo augment two matrices, follow these steps:\n<ol class=\”level-one\”>\n <li>To select the Augment command from the MATRX MATH menu, press\n<img src=\”https://www.dummies.com/wp-content/uploads/400128.image4.jpg\” width=\”116\” height=\”33\” alt=\”image4.jpg\”/>\n </li>\n <li>Enter the first matrix and then press [,] (see the first screen).\nTo create a matrix from scratch, press [ALPHA][ZOOM]. To access a stored matrix, press [2nd][x–1].\n </li>\n <li>Enter the second matrix and then press [ENTER].\nThe second screen displays the augmented matrix.\n </li>\n <li>Store your augmented matrix by pressing\n<img src=\”https://www.dummies.com/wp-content/uploads/400129.image5.jpg\” width=\”206\” height=\”33\” alt=\”image5.jpg\”/>\nThe augmented matrix is stored as [C]. See the third screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400130.image6.jpg\” width=\”535\” height=\”161\” alt=\”image6.jpg\”/>\n </li>\n</ol>\nSystems of linear equations can be solved by first putting the augmented matrix for the system in reduced row-echelon form. The mathematical definition of reduced row-echelon form isn’t important here. It’s simply an equivalent form of the original system of equations, which, when converted back to a system of equations, gives you the solutions (if any) to the original system of equations.\nTo find the reduced row-echelon form of a matrix, follow these steps:\n<ol class=\”level-one\”>\n <li>To scroll to the rref( function in the MATRX MATH menu, press\n<img src=\”https://www.dummies.com/wp-content/uploads/400131.image7.jpg\” width=\”96\” height=\”33\” alt=\”image7.jpg\”/>\nand use the up-arrow key. See the first screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400132.image8.jpg\” width=\”535\” height=\”244\” alt=\”image8.jpg\”/>\n </li>\n <li>Press [ENTER] to paste the function on the Home screen.\n </li>\n <li>Press [2nd][x–1] and press [3] to choose the augmented matrix you just stored.\n </li>\n <li>Press [ENTER] to find the solution.\nSee the second screen.\n </li>\n</ol>\nTo find the solutions (if any) to the original system of equations, convert the reduced row-echelon matrix to a system of equations: \n<img src=\”https://www.dummies.com/wp-content/uploads/400133.image9.jpg\” width=\”104\” height=\”64\” alt=\”image9.jpg\”/>\nAs you see, the solutions to the system are x = 5, y = 0, and z = 1. Unfortunately, not all systems of equations have unique solutions like this system. Here are examples of the two other cases that you may see when solving systems of equations:\n<img src=\”https://www.dummies.com/wp-content/uploads/400134.image10.jpg\” width=\”236\” height=\”65\” alt=\”image10.jpg\”/>\nSee the reduced row-echelon matrix solutions to the preceding systems in the first two screens.\n<img src=\”https://www.dummies.com/wp-content/uploads/400135.image11.jpg\” width=\”535\” height=\”242\” alt=\”image11.jpg\”/>\nTo find the solutions (if any), convert the reduced row-echelon matrices to a system of equations:\n<img src=\”https://www.dummies.com/wp-content/uploads/400136.image12.jpg\” width=\”248\” height=\”65\” alt=\”image12.jpg\”/>\nBecause one of the equations in the first system simplifies to 0 = 1, this system has no solution. In the second system, one of the equations simplifies to 0 = 0. This indicates the system has an infinite number of solutions that are on the line x + 6y = 10.”,”description”:”Matrices are the perfect tool for solving systems of equations (the larger the better). Fortunately, you can work with matrices on your TI-84 Plus. All you need to do is decide which method you want to use. \n<h2 id=\”tab1\” >A–1*B method of solving a system of equations</h2>\nWhat do the A and B represent? The letters A and B are capitalized because they refer to matrices. Specifically, A is the coefficient matrix and B is the constant matrix. In addition, X is the variable matrix. No matter which method you use, it’s important to be able to convert back and forth from a system of equations to matrix form.\n<img src=\”https://www.dummies.com/wp-content/uploads/400124.image0.jpg\” width=\”110\” height=\”64\” alt=\”image0.jpg\”/>\n<img src=\”https://www.dummies.com/wp-content/uploads/400125.image1.jpg\” width=\”487\” height=\”357\” alt=\”image1.jpg\”/>\nHere’s a short explanation of where this method comes from. Any system of equations can be written as the matrix equation, A * X = B. By pre-multiplying each side of the equation by A–1 and simplifying, you get the equation X = A–1 * B. \nUsing your calculator to find A–1 * B is a piece of cake. Just follow these steps:\n<ol class=\”level-one\”>\n <li>Enter the coefficient matrix, A.\nPress [ALPHA][ZOOM] to create a matrix from scratch or press [2nd][x–1] to access a stored matrix. See the first screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400126.image2.jpg\” width=\”535\” height=\”160\” alt=\”image2.jpg\”/>\n </li>\n <li>Press [x–1] to find the inverse of matrix A.\nSee the second screen.\n </li>\n <li>Enter the constant matrix, B.\n </li>\n <li>Press [ENTER] to evaluate the variable matrix, X.\nThe variable matrix indicates the solutions: x = 5, y = 0, and z = 1. See the third screen.\n </li>\n</ol>\nIf the determinant of matrix A is zero, you get the ERROR: SINGULAR MATRIX error message. This means that the system of equations has either no solution or infinite solutions.\n<h2 id=\”tab2\” >Augmenting matrices method to solve a system of equations</h2>\nAugmenting two matrices enables you to append one matrix to another matrix. Both matrices must be defined and have the same number of rows. Use the system of equations to augment the coefficient matrix and the constant matrix.\n<img src=\”https://www.dummies.com/wp-content/uploads/400127.image3.jpg\” width=\”110\” height=\”64\” alt=\”image3.jpg\”/>\nTo augment two matrices, follow these steps:\n<ol class=\”level-one\”>\n <li>To select the Augment command from the MATRX MATH menu, press\n<img src=\”https://www.dummies.com/wp-content/uploads/400128.image4.jpg\” width=\”116\” height=\”33\” alt=\”image4.jpg\”/>\n </li>\n <li>Enter the first matrix and then press [,] (see the first screen).\nTo create a matrix from scratch, press [ALPHA][ZOOM]. To access a stored matrix, press [2nd][x–1].\n </li>\n <li>Enter the second matrix and then press [ENTER].\nThe second screen displays the augmented matrix.\n </li>\n <li>Store your augmented matrix by pressing\n<img src=\”https://www.dummies.com/wp-content/uploads/400129.image5.jpg\” width=\”206\” height=\”33\” alt=\”image5.jpg\”/>\nThe augmented matrix is stored as [C]. See the third screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400130.image6.jpg\” width=\”535\” height=\”161\” alt=\”image6.jpg\”/>\n </li>\n</ol>\nSystems of linear equations can be solved by first putting the augmented matrix for the system in reduced row-echelon form. The mathematical definition of reduced row-echelon form isn’t important here. It’s simply an equivalent form of the original system of equations, which, when converted back to a system of equations, gives you the solutions (if any) to the original system of equations.\nTo find the reduced row-echelon form of a matrix, follow these steps:\n<ol class=\”level-one\”>\n <li>To scroll to the rref( function in the MATRX MATH menu, press\n<img src=\”https://www.dummies.com/wp-content/uploads/400131.image7.jpg\” width=\”96\” height=\”33\” alt=\”image7.jpg\”/>\nand use the up-arrow key. See the first screen.\n<img src=\”https://www.dummies.com/wp-content/uploads/400132.image8.jpg\” width=\”535\” height=\”244\” alt=\”image8.jpg\”/>\n </li>\n <li>Press [ENTER] to paste the function on the Home screen.\n </li>\n <li>Press [2nd][x–1] and press [3] to choose the augmented matrix you just stored.\n </li>\n <li>Press [ENTER] to find the solution.\nSee the second screen.\n </li>\n</ol>\nTo find the solutions (if any) to the original system of equations, convert the reduced row-echelon matrix to a system of equations: \n<img src=\”https://www.dummies.com/wp-content/uploads/400133.image9.jpg\” width=\”104\” height=\”64\” alt=\”image9.jpg\”/>\nAs you see, the solutions to the system are x = 5, y = 0, and z = 1. Unfortunately, not all systems of equations have unique solutions like this system. Here are examples of the two other cases that you may see when solving systems of equations:\n<img src=\”https://www.dummies.com/wp-content/uploads/400134.image10.jpg\” width=\”236\” height=\”65\” alt=\”image10.jpg\”/>\nSee the reduced row-echelon matrix solutions to the preceding systems in the first two screens.\n<img src=\”https://www.dummies.com/wp-content/uploads/400135.image11.jpg\” width=\”535\” height=\”242\” alt=\”image11.jpg\”/>\nTo find the solutions (if any), convert the reduced row-echelon matrices to a system of equations:\n<img src=\”https://www.dummies.com/wp-content/uploads/400136.image12.jpg\” width=\”248\” height=\”65\” alt=\”image12.jpg\”/>\nBecause one of the equations in the first system simplifies to 0 = 1, this system has no solution. In the second system, one of the equations simplifies to 0 = 0. This indicates the system has an infinite number of solutions that are on the line x + 6y = 10.”,”blurb”:””,”authors”:[{“authorId”:9554,”name”:”Jeff McCalla”,”slug”:”jeff-mccalla”,”description”:” Jeff McCalla is a mathematics teacher at St. Mary’s Episcopal School in Memphis, TN. He cofounded the TI-Nspire SuperUser group, and received the Presidential Award for Excellence in Science & Mathematics Teaching. C.C. Edwards is an educator who has presented numerous workshops on using TI calculators.”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9554″}},{“authorId”:9555,”name”:”C. C. Edwards”,”slug”:”c-c-edwards”,”description”:” Jeff McCalla is a mathematics teacher at St. Mary’s Episcopal School in Memphis, TN. He cofounded the TI-Nspire SuperUser group, and received the Presidential Award for Excellence in Science & Mathematics Teaching. C.C. Edwards is an educator who has presented numerous workshops on using TI calculators.”,”hasArticle”:false,”_links”:{“self”:”https://dummies-api.dummies.com/v2/authors/9555″}}],”primaryCategoryTaxonomy”:{“categoryId”:33551,”title”:”Graphing Calculators”,”slug”:”graphing-calculators”,”_links”:{“self”:”https://dummies-api.dummies.com/v2/categories/33551″}},”secondaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”tertiaryCategoryTaxonomy”:{“categoryId”:0,”title”:null,”slug”:null,”_links”:null},”trendingArticles”:null,”inThisArticle”:[{“label”:”A–1*B method of solving a system of equations”,”target”:”#tab1″},{“label”:”Augmenting matrices method to solve a system of equations”,”target”:”#tab2″}],”relatedArticles”:{“fromBook”:[{“articleId”:294935,”title”:”TI-84 Plus CE Graphing Calculator For Dummies Cheat Sheet”,”slug”:”ti-84-plus-ce-graphing-calculator-for-dummies-cheat-sheet”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/294935″}},{“articleId”:257053,”title”:”How to Find Standard Deviation on the TI-84 Graphing Calculator”,”slug”:”how-to-find-standard-deviation-on-the-ti-84-graphing-calculator”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/257053″}},{“articleId”:209964,”title”:”How to Enable and Disable the TI-TestGuard App on a Class Set of TI-84 Plus Calculators”,”slug”:”how-to-enable-and-disable-the-ti-testguard-app-on-a-class-set-of-ti-84-plus-calculators”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209964″}},{“articleId”:209963,”title”:”How to Download and Install the TI-TestGuard App on the TI-84 Plus”,”slug”:”how-to-download-and-install-the-ti-testguard-app-on-the-ti-84-plus”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209963″}},{“articleId”:209962,”title”:”How to Store a Picture on the TI-84 Plus”,”slug”:”how-to-store-a-picture-on-the-ti-84-plus”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209962″}}],”fromCategory”:[{“articleId”:294935,”title”:”TI-84 Plus CE Graphing Calculator For Dummies Cheat Sheet”,”slug”:”ti-84-plus-ce-graphing-calculator-for-dummies-cheat-sheet”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/294935″}},{“articleId”:257053,”title”:”How to Find Standard Deviation on the TI-84 Graphing Calculator”,”slug”:”how-to-find-standard-deviation-on-the-ti-84-graphing-calculator”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/257053″}},{“articleId”:209964,”title”:”How to Enable and Disable the TI-TestGuard App on a Class Set of TI-84 Plus Calculators”,”slug”:”how-to-enable-and-disable-the-ti-testguard-app-on-a-class-set-of-ti-84-plus-calculators”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209964″}},{“articleId”:209962,”title”:”How to Store a Picture on the TI-84 Plus”,”slug”:”how-to-store-a-picture-on-the-ti-84-plus”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209962″}},{“articleId”:209963,”title”:”How to Download and Install the TI-TestGuard App on the TI-84 Plus”,”slug”:”how-to-download-and-install-the-ti-testguard-app-on-the-ti-84-plus”,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”_links”:{“self”:”https://dummies-api.dummies.com/v2/articles/209963″}}]},”hasRelatedBookFromSearch”:false,”relatedBook”:{“bookId”:281880,”slug”:”ti-84-plus-graphing-calculator-for-dummies-2nd-edition”,”isbn”:”9781119887607″,”categoryList”:[“technology”,”electronics”,”graphing-calculators”],”amazon”:{“default”:”https://www.amazon.com/gp/product/1119887607/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”ca”:”https://www.amazon.ca/gp/product/1119887607/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”indigo_ca”:”http://www.tkqlhce.com/click-9208661-13710633?url=https://www.chapters.indigo.ca/en-ca/books/product/1119887607-item.html&cjsku=978111945484″,”gb”:”https://www.amazon.co.uk/gp/product/1119887607/ref=as_li_tl?ie=UTF8&tag=wiley01-20″,”de”:”https://www.amazon.de/gp/product/1119887607/ref=as_li_tl?ie=UTF8&tag=wiley01-20″},”image”:{“src”:null,”width”:0,”height”:0},”title”:”TI-84 Plus CE Graphing Calculator For Dummies, 3rd Edition”,”testBankPinActivationLink”:””,”bookOutOfPrint”:true,”authorsInfo”:”Jeff McCalla is a mathematics teacher at St. Mary’s Episcopal School in Memphis, TN. 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How to Solve a System of Equations on the TI-84 Plus

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